On the structure of the orthotropic 3D permeability tensor of an anisotropic porous body in heat and mass transfer problems

Abstract

The structure of a 3D permeability tensor with orthotropic properties is proposed to describe various heat and mass transfer processes in anisotropic porous media. This structure of the permeability tensor has advantages over the monoclinic and triclinic because of the minimum number of elements to be determined, which greatly simplifies the formulation, conduct and processing of the corresponding experiments. In addition, the confirmation of the operability of such a structure opens up new opportunities for the artificial creation of porous materials with specified properties and the required anisotropy architecture on 3D printers. Using the Jacobians of rotations in the Cartesian reference system, the matrix of rotations is found as a result of successive multiplication of Jacobians by different angles around the axes of the base coordinate system. The final form of the tensor is identified as the product of the orthotropic tensor on the left and right by the turn matrix and the transposed turn matrix, respectively. This representation allowed us to calculate the inverse permeability tensor and synthesize a mathematical 3D model of Darcy-Brinkman type of unidirectional flow in a porous channel of rectangular cross-section on the example of hydrodynamic filtration of a Newtonian fluid. It is shown that such linearization does not level the property of anisotropy in a porous medium, while maintaining three-dimensionality.